Fast Frequency Estimation by Zero Crossings of Differential Spline Wavelet Transform

Abstract

Zero crossings or extrema of a wavelet transform constitute important signatures for signal analysis with the advantage of great simplicity. In this paper, we introduce a fast frequency-estimation method based on zero-crossing counting in the transform domain of a family of differential spline wavelets. The resolution and order of the vanishing moments of the chosen wavelets have a close relation with the frequency components of a signal. Theoretical results on estimating the highest and the lowest frequency components are derived, which are particularly useful for frequency estimation of harmonic signals. The results are illustrated with the help of several numerical examples. Finally, we discuss the connection of this approach with other frequency estimation methods, with the high-order level-crossing analysis in statistics, and with the scaling theorem in computer vision.

Fast frequency estimation using spline wavelets

“…The proof can be found in. 10 Theorem 4.1. For a zero mean stationary Gaussian random signal {X t }, t = 1, · · · , N, let ω be the lowest frequency in the spectrum.…”

“…Following the same process, we can obtain similar results regarding the effect on the estimation of spectrum using derivative filters of different orders. 10 Theorem 4.2. For a zero mean stationary Gaussian random signal {X t }, t = 1, · · · , N, let ω be the highest frequency in the spectrum.…”